GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 21 Mar 2019, 13:20

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# If M, N are integers greater than 1 such that 2M<N, which of the foll

Author Message
TAGS:

### Hide Tags

GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 906
If M, N are integers greater than 1 such that 2M<N, which of the foll  [#permalink]

### Show Tags

18 Feb 2019, 09:10
00:00

Difficulty:

95% (hard)

Question Stats:

30% (02:24) correct 70% (02:50) wrong based on 50 sessions

### HideShow timer Statistics

GMATH practice exercise (Quant Class 17)

If M, N are integers greater than 1 such that 2M<N, which of the following numbers could be twice the value of the sum of all integers from M to N, including both of them?

I. 54
II. 52
III. 50

(A) I. only
(B) II. only
(C) III. only
(D) Exactly two of them
(E) None of them

The tricky alternative choice would be "All of them". We believe the examiner would NOT put it among the answer choices!

_________________

Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net

VP
Joined: 09 Mar 2018
Posts: 1007
Location: India
Re: If M, N are integers greater than 1 such that 2M<N, which of the foll  [#permalink]

### Show Tags

18 Feb 2019, 09:29
fskilnik wrote:
GMATH practice exercise (Quant Class 17)

If M, N are integers greater than 1 such that 2M<N, which of the following numbers could be twice the value of the sum of all integers from M to N, including both of them?

I. 54
II. 52
III. 50

(A) I. only
(B) II. only
(C) III. only
(D) Exactly two of them
(E) None of them

Would still keep this, though brute force is a time consuming approach.

2M<N, N >2M, one thing to observe was that, the series will always start from an even Integer

Will start from a single digit, 8,9,10,11,12 => 50

Now we can think greater values now

12,13,14,15 => 54

D
_________________

If you notice any discrepancy in my reasoning, please let me know. Lets improve together.

Quote which i can relate to.
Many of life's failures happen with people who do not realize how close they were to success when they gave up.

Senior Manager
Joined: 04 Aug 2010
Posts: 383
Schools: Dartmouth College
If M, N are integers greater than 1 such that 2M<N, which of the foll  [#permalink]

### Show Tags

18 Feb 2019, 18:25
1
fskilnik wrote:
GMATH practice exercise (Quant Class 17)

If M, N are integers greater than 1 such that 2M<N, which of the following numbers could be twice the value of the sum of all integers from M to N, including both of them?

I. 54
II. 52
III. 50

(A) I. only
(B) II. only
(C) III. only
(D) Exactly two of them
(E) None of them

54, 52 and 50 are options for TWICE the sum.
Thus, I, II, and III imply the following options for the ACTUAL SUM:
I: 27
II: 26
III: 25

Case 1: M=2
Since it is required that N>2M, N≥5
M=2 and N=5 --> sum = 2+3+4+5 = 14
M=2 and N=6 --> sum increases by 6 --> 14+6 = 20
M=2 and N=7 --> sum increases by 7 --> 20+7 = 27
The green case indicates that M=2 and N=7 will yield a sum of 27, implying that option I -- 54 -- can be the value of TWICE the sum.

Case 2: M=3
Since it is required that N>2M, N≥7
M=3 and N=7 --> sum = 3+4+5+6+7 = 25
The green case indicates that M=3 and N=7 will yield a sum of 25, implying that option III -- 50 -- can be the value of TWICE the sum.

Thus, I and III are possible.

_________________

GMAT and GRE Tutor
Over 1800 followers
GMATGuruNY@gmail.com
New York, NY
If you find one of my posts helpful, please take a moment to click on the "Kudos" icon.
Available for tutoring in NYC and long-distance.

Intern
Joined: 28 Jan 2019
Posts: 10
Re: If M, N are integers greater than 1 such that 2M<N, which of the foll  [#permalink]

### Show Tags

19 Feb 2019, 00:04
1
Hi, this is my first answer. Kindly check and suggest if my approach is right.

Given, 2M<N => Assuming the AP has only 2 numbers in the series, M and N, M+N>3M.
Thus solving 3M with the options (54/2, 52/2, 50/2) should give a reminder, suggesting that the sum of M+N is greater than the values.

That is,
3M=27, No remainder. Therefore, M+N not greater than 3M.
3M=26, Remainder = 2. Therefore, M+N>3M.
3M=25, Remainder = 1. Therefore, M+N>3M.

Hence, (ii) and (iii) could be the answers, hence D.
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 906
Re: If M, N are integers greater than 1 such that 2M<N, which of the foll  [#permalink]

### Show Tags

19 Feb 2019, 07:13
OhsostudiousMJ wrote:
Hi, this is my first answer. Kindly check and suggest if my approach is right.

Given, 2M<N => Assuming the AP has only 2 numbers in the series, M and N, M+N>3M.
Thus solving 3M with the options (54/2, 52/2, 50/2) should give a reminder, suggesting that the sum of M+N is greater than the values.

That is,
3M=27, No remainder. Therefore, M+N not greater than 3M.
3M=26, Remainder = 2. Therefore, M+N>3M.
3M=25, Remainder = 1. Therefore, M+N>3M.

Hence, (ii) and (iii) could be the answers, hence D.

Hi OhsostudiousMJ ,

Welcome to our GMAT Club community!

You have discussed whether M+N is greater than 3M, but this is not our focus. This problem is hard, even in terms of understanding what is going on...

Suggestion: start dealing with easier questions (600-700 or below 600). You can find each question´s level looking at the questions tags when they are posted.

Regards and success in your studies!
Fabio.
_________________

Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net

GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 906
If M, N are integers greater than 1 such that 2M<N, which of the foll  [#permalink]

### Show Tags

19 Feb 2019, 07:40
fskilnik wrote:
GMATH practice exercise (Quant Class 17)

If M, N are integers greater than 1 such that 2M<N, which of the following numbers could be twice the value of the sum of all integers from M to N, including both of them?

I. 54
II. 52
III. 50

(A) I. only
(B) II. only
(C) III. only
(D) Exactly two of them
(E) None of them

Although the problem seems hard, the focused-numbers (half the numbers given) are SMALL... and that´s the hint to try the "organized manual work" (chosen in previous solutions)!

We are looking for the 25 (III), 26 (II) and 27 (I) possibilities.

$${S_K} = 1 + 2 + \ldots + K = {{K\left( {K + 1} \right)} \over 2}\,\,\,\,\,\,\,\,\left( {{\rm{arithmetic}}\,\,{\rm{sequence}}} \right)$$

$${S_7} = 7 \cdot 4 = 28\,\,\, \Rightarrow \,\,\,\left\{ \matrix{ \,27 = 28 - 1 = \left( {1 + 2 + \ldots + 7} \right) - 1\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {N,M} \right) = \left( {7,2} \right)\,\,\,{\rm{with}}\,\,N > 2M\,\,\, \Rightarrow \,\,\,\,{\rm{viable!}} \hfill \cr \,25 = 28 - 3 = \left( {1 + 2 + \ldots + 7} \right) - \left( {1 + 2} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {N,M} \right) = \left( {7,3} \right)\,\,\,{\rm{with}}\,\,N > 2M\,\,\, \Rightarrow \,\,\,\,{\rm{viable!}} \hfill \cr} \right.$$

From the alternative choices given, we are sure we have found (EASILY!) the right answer!

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.

POST-MORTEM:

Question 1.: how can we be sure that 26 (II) cannot be obtained? Please note that:

$$\left. \matrix{ {S_8} = 4 \cdot 9 = 36 \hfill \cr {S_4} = 2 \cdot 10 = 10\,\,\, \hfill \cr} \right\}\,\,\, \Rightarrow \,\,\,26 = 36 - 10 = \left( {1 + 2 + \ldots + 7 + 8} \right) - \left( {1 + 2 + 3 + 4} \right)$$

and the ONLY reason (N,M) = (8,5) must be refuted is the fact that N>2M is false... What about other values for (N,M)? How can we be SURE there is not a single viable possibility?

Question 2.: which mathematical (GMAT-focused) properties could be useful to find all possible (N,M) pairs for a given LARGER value?

We will address both questions in our very next problem, here: https://gmatclub.com/forum/a-number-is- ... l#p2229235
_________________

Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net

If M, N are integers greater than 1 such that 2M<N, which of the foll   [#permalink] 19 Feb 2019, 07:40
Display posts from previous: Sort by